Optimal boundary control of a simplified Ericksen--Leslie system for nematic liquid crystal flows in $2D$
Abstract
In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen--Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier--Stokes equations for the fluid velocity coupled with a convective Ginzburg--Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fr\'echet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.
Keywords
Cite
@article{arxiv.1609.03861,
title = {Optimal boundary control of a simplified Ericksen--Leslie system for nematic liquid crystal flows in $2D$},
author = {C. Cavaterra and E. Rocca and H. Wu},
journal= {arXiv preprint arXiv:1609.03861},
year = {2023}
}